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Mirrors > Home > MPE Home > Th. List > nfielex | Structured version Visualization version GIF version |
Description: If a class is not finite, then it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.) |
Ref | Expression |
---|---|
nfielex | ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0fi 9081 | . . . 4 ⊢ ∅ ∈ Fin | |
2 | eleq1 2827 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ Fin ↔ ∅ ∈ Fin)) | |
3 | 1, 2 | mpbiri 258 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ Fin) |
4 | 3 | con3i 154 | . 2 ⊢ (¬ 𝐴 ∈ Fin → ¬ 𝐴 = ∅) |
5 | neq0 4358 | . 2 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | sylib 218 | 1 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∅c0 4339 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-om 7888 df-en 8985 df-fin 8988 |
This theorem is referenced by: cusgrfi 29491 esumcst 34044 topdifinffinlem 37330 |
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